Problem 34 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=-5 \sqrt{x} $$

Step-by-Step Solution

Answer

Reflect and stretch \( y=\sqrt{x} \) to draw \( y=-5\sqrt{x} \).

Step 1: Understand the Standard Function

The given function is a transformation of the square root function, which is represented as \( y = \sqrt{x} \). This function is a basic function that starts at the origin (0,0) and extends to the right, increasing slowly.

Step 2: Apply the Vertical Stretch and Reflection

The function \( y = -5\sqrt{x} \) indicates two transformations on \( y = \sqrt{x} \). First, there is a vertical stretch by a factor of 5, which means each point on \( y = \sqrt{x} \) is moved 5 times further away from the x-axis. Second, there is a reflection across the x-axis due to the negative sign. This reflects the vertically stretched graph downward.

Step 3: Draw the Transformed Function

Starting with the basic shape of \( y = \sqrt{x} \), apply the transformations. Reflect the initial curve across the x-axis and stretch it vertically by multiplying the \( y \)-values by 5. The resulting graph will start from the origin and curve downwards.

Key Concepts

Graph Sketching

Sketching a graph is a way of visualizing a function without relying solely on plotted points. This approach helps us to understand the overall shape and transformations applied to a function. When sketching the graph of a transformed function, start with the graph of a standard or "parent" function. From there, apply the relevant transformations to accurately depict how the alterations affect the function's behavior and appearance. Here, we focus on the square root function and its transformations.

Vertical Stretch

A vertical stretch occurs when the output values (or y-values) of a function are multiplied by a constant factor. This transformation affects how stretched or compressed a graph appears vertically, without altering the x-values.
For instance, in the function

• The \( y = -5\sqrt{x} \) undergoes a vertical stretch by a factor of 5.
• This transformation takes the outputs (y-values) and multiplies them by 5.
• As a result, every point on the original \( y = \sqrt{x} \) graph moves 5 times further from the x-axis, making the graph steeper.

Understanding this transformation is crucial to accurately sketch the graph of the function.

Reflection Across Axis

A reflection across an axis involves flipping a graph over a specified line. For our function, reflecting across the x-axis takes place. Reflection in this way changes the sign of the y-values, causing the graph to invert.

• In our function: the presence of a negative sign before the expression \( y = -5\sqrt{x} \) indicates a reflection over the x-axis.
• Essentially, the entire graph of \( y = 5\sqrt{x} \) flips downwards, creating a mirror image across the horizontal axis.

Understanding reflection—including how and where it applies—helps in accurately sketching the function's modification. This also affects how we subsequently interpret the graph visually.

Square Root Function

The square root function \( y = \sqrt{x} \) serves as the basic foundation in understanding function transformations. Starting at the origin (0,0), the graph of \( y = \sqrt{x} \) extends to the right along the positive x-axis. It increases gradually, forming a curve.

• It is important to know the behavior of this parent function because it serves as the base model before any alterations.
• Understanding \( y = \sqrt{x} \)'s shape and direction allows for more straightforward application of any additional transformations, such as stretches or reflections.

Knowing the basic square root function lets you better visualize the transformed function. This is essential for both graph sketching and interpreting transformations successfully.